SKEW NEAR-LATTICES IN RICKART RINGS J¯ anis C ¯ ırulis University of Latvia email: jc@lanet.lv 88. Arbeitstagung Allgemeine Algebra (AAA88) Warszawa, June 19 - 22, 2014
OVERVIEW 1. Basic definitions and facts (skew near-lattices, examples, Rickart rings) 2. Rickart rings as skew near-lattices 1
1. BASIC DEFINITIONS AND FACTS 1.1 Skew nearlattices [J. C ¯ ırulis, Skew nearlattices: some structure and representation theorems , Contrib. General Algebra 19 (2010), 33–44] The natural order ≤ in a band ( A, △ ) is defined by x ≤ y iff y △ x = x = x △ y . A skew nearlattice is a band A which has the upper bound prop- erty under the natural ordering. Let ∨ and ∧ stand for the partial lattice operations in A under ≤ . 2
A particular class of skew nearlattices is that in which △ is com- mutative in every initial segment of A . This is the case if and only if the underlying band is normal , i.e., satisfies the identity: u △ x △ y △ v = u △ y △ x △ v , and then every initial segment [0 , p ] is a lattice with a ∧ b = a △ b . 3
(Recall: x ≤ y iff y △ x = x = x △ y .) The skew nearlattices arising in some applications are right- (or left- ) handed in the following sense: x = x △ y implies that y △ x = x (i.e., if x ≤ y iff x △ y = x ). This is the case if and only if the underlying band is right regular , i.e., satisfies the identity x △ y △ x = y △ x . A normal right-handed band is right normal (and conversely), i.e., satisfies the identity x △ y △ z = y △ x △ z . 4
1.2. Three examples of right normal skew nearlattices � The set of partial functions X → V with △ defined by φ △ ψ := ψ | (dom φ ∩ dom ψ ). � The set of all subsets of X × Y (i.e., binary relations) with △ defined by R △ S := S ◦ ( I R ∩ I S ), where ◦ is composition, I R is the identity relation restricted to the range of R , and likewise I S . � The set of self-adjoint operators of a Hilbert space H with △ defined by A △ B := B max( P ∈ P : P ≤ P A ∧ P B , BP = PB ), where P is the lattice of projection operators in H , and P C is the projection operator onto the closed range of C . [ A △ B := B ( P A ∧ P B ) for all bounded operators ] 5
1.3. Rickart rings A (right) Rickart ring is an associative ring R such that the right annihilator of every element of R is a principal right ideal generated by an idempotent. A Rickart *-ring is an involution ring R which is Rickart and every each generating idempotent in the previous definition is symmetric (alias self-adjoint). We shall deal with Rickart rings which have certain involution- free properties of Rickart *-rings without introducing any invo- lution. 6
Let R be a Rickart ring. This means that given x ∈ R , we can choose an idempotent x ′ such that ✎ ☞ for all y ∈ R , xy = 0 iff x ′ y = y . ✍ ✌ We may assume that here ☛ ✟ x ′′ = 1 − x ′ , where 1 := 0 ′ . 1 is the unit of R ✡ ✠ This identity is equivalent to the condition ✎ ☞ for all y ∈ R , xy = 0 iff x ′′ y = 0. ✍ ✌ Idempotents of R in the range of the operation ′ are said to be closed . We denote the set of all such idempotents by P . Rickart rings, considered as algebras ( R, + , · , ′ , 0), form a variety. 7
Recall that idempotents of any ring form an orthomodular poset under the ordering e ≤ f iff fe = e = ef . We call a right Rickart ring strong if e ≤ f iff ef = e for all closed idempotents e and f . Proposition If R is strong, then P is an orthomodular lattice. In particular, every initial segment [0 , g ] of P is � a sublattice of P , � orthomodular, with orthocomplementation ⊥ g given by e ⊥ g := g − e . 8
2. A RICKART RING AS A SKEW NEARLATTICE R – a strong Rickart ring. Let x △ y := y ( x ′′ ∧ y ′′ ) (the skew meet of x an y ). Theorem 1 (a) The algebra ( R, △ ) is a right normal skew nearlattice with 0 the least element. (b) The partial join operation ∨ in R is given by if a, b ≤ x , then a ∨ b = x ( a ′′ ∨ b ′′ ). (c) The lattice [0 , 1] coincides with P . If the ring R happens to be regular, then the order ≤ agrees with the so called right star order defined by x ≤∗ y iff x = yx ′′ and Rx ⊆ Ry . 9
Theorem 2 (a) The mapping φ : x �→ x ′′ is an order homomorfism of R onto P ; moreover, φ ( x △ y ) = φ ( x ) ∧ φ ( y ). (b) For every x ∈ R , the restriction of φ to [0 , x ] is an order isomorphism between this segment and [0 , x ′′ ]. (c) The kernel equivalence of φ is the Green’s equivalence R (and even D ) of the band ( R, △ ). Corollary (a) Every initial segment of R is an orthomodular lattice. (b) If x R y , then the segments [0 , x ] and [0 , y ] are order iso- morphic, the isomorphism being given by a �→ ya ′′ . ( a ∈ [0 , x ]) (c) Conversely, if these segments are order isomorphic, then x R y . 10
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